Optimal. Leaf size=240 \[ \frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}+\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a f \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.27, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3196, 474, 583, 524, 426, 424, 421, 419} \[ \frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}+\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rule 424
Rule 426
Rule 474
Rule 524
Rule 583
Rule 3196
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{x^4 \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {-2 (2 a+b)+(3 a+b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a f}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {-a (3 a+b)-2 b (2 a+b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 f}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {\left ((a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a f}+\frac {\left (2 (2 a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 f}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {\left (2 (2 a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left ((a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 a f \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 3.97, size = 186, normalized size = 0.78 \[ \frac {\frac {\cot (e+f x) \csc ^2(e+f x) \left ((2 a+b) (2 a+b \cos (4 (e+f x))+3 b)-2 \left (4 a^2+5 a b+2 b^2\right ) \cos (2 (e+f x))\right )}{\sqrt {2}}-2 a (a+b) \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac {b}{a}\right .\right )+4 a (2 a+b) \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{6 a^2 f \sqrt {2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \cot \left (f x + e\right )^{4}}{b \cos \left (f x + e\right )^{2} - a - b}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (f x + e\right )^{4}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.96, size = 351, normalized size = 1.46 \[ -\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )+b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \left (\sin ^{3}\left (f x +e \right )\right )-4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \left (\sin ^{3}\left (f x +e \right )\right )+4 a b \left (\sin ^{6}\left (f x +e \right )\right )+2 b^{2} \left (\sin ^{6}\left (f x +e \right )\right )+4 a^{2} \left (\sin ^{4}\left (f x +e \right )\right )-3 a b \left (\sin ^{4}\left (f x +e \right )\right )-2 b^{2} \left (\sin ^{4}\left (f x +e \right )\right )-5 a^{2} \left (\sin ^{2}\left (f x +e \right )\right )-a b \left (\sin ^{2}\left (f x +e \right )\right )+a^{2}}{3 a^{2} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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